What kind of damping is this $F = -ax|x'|$? From Applied Mathematics by Logan:

A mass hanging on a spring is <...> governed by
  $$mx'' = -ax|x'| - kx$$
  where $-ax|x'|$ is a nonlinear damping force.

I looked up "nonlinear damping" and found $F=-a|x'|^px'$ which is clear enough.
As far as I know one thinks of $F= -ax'$ damping as of viscous drag.
What is the interpretation of $F = -ax|x'|$ damping?
It confuses me that it has $x$ in it. Why would the damping force care whether the mass is above or below the origin? Shouldn't the damping force depend solely on the velocity of the mass?
 A: I would look at this in a slightly different way. Rearranging it:
$$ m \ddot{x} = -(a|\dot{x}|+k) x = -k_{eff} x$$
If you look at it that way, it is really a variable, non-linear stiffness $k_{eff}$ that depends on the velocity, rather than a damping that depends on the position. In this respect (assuming $a > 0$), the stiffness coefficient has a lower bound of $k$ and the maximum depends on the maximum velocity. Given a periodic-type motion, at least initially, the stiffness would be $k$ at $\pm x_{max}$ and the stiffness would be a maximum when it was near $x = 0$, where velocity is a maximum. 
Given all of that, I cannot think of exactly what kind of physical scenario this would represent off the top of my head. But one could imagine it is a mass oscillating somehow in a non-Newtonian fluid. I suppose it would be a shear-thickening fluid where viscosity is zero when the body is not moving and viscosity is a maximum when the body is moving fastest. 
Displacement-dependent dampers seem to have applications in automotive and aircraft suspensions -- see this paper and references at the end of the introduction. It would appear that hydraulic systems pushing/pulling fluid through an orifice may be modeled this way, as well as tapered pistons and a few other things. 
